Pseudolikelihood

In statistical theory, a pseudolikelihood is an approximation to the joint probability distribution of a collection of random variables. The practical use of this is that it can provide an approximation to the likelihood function of a set of observed data which may either provide a computationally simpler problem for estimation, or may provide a way of obtaining explicit estimates of model parameters. The pseudolikelihood approach was introduced by Julian Besag in the context of analysing data having spatial dependence.

In statistical theory, a pseudolikelihood is an approximation to the joint probability distribution of a collection of random variables. The practical use of this is that it can provide an approximation to the likelihood function of a set of observed data which may either provide a computationally simpler problem for estimation, or may provide a way of obtaining explicit estimates of model parameters. The pseudolikelihood approach was introduced by Julian Besag in the context of analysing data having spatial dependence.

Definition

Given a set of random variables X = X_1, X_2, \ldots, X_n the pseudolikelihood of X = x = (x_1,x_2, \ldots, x_n) is L(\theta) := \prod_i \mathrm{Pr}_\theta(X_i = x_i\mid X_j = x_j \text{ for } j \neq i)=\prod_i \mathrm {Pr}_\theta (X_i = x_i \mid X_{-i}=x_{-i}) in discrete case and L(\theta) := \prod_i p_\theta(x_i \mid x_j \text{ for } j \neq i)=\prod_i p _\theta (x_i \mid x_{-i})=\prod _i p_\theta (x_i \mid x_1,\ldots, \hat x_i, \ldots, x_n) in continuous one. Here X is a vector of variables, x is a vector of values, p_\theta(\cdot \mid \cdot) is conditional density and \theta =(\theta_1, \ldots, \theta_p) is the vector of parameters we are to estimate. The expression X = x above means that each variable X_i in the vector X has a corresponding value x_i in the vector x and x_{-i}=(x_1, \ldots,\hat x_i, \ldots, x_n) means that the coordinate x_i has been omitted. The expression \mathrm {Pr}_\theta(X = x) is the probability that the vector of variables X has values equal to the vector x . This probability of course depends on the unknown parameter \theta . Because situations can often be described using state variables ranging over a set of possible values, the expression \mathrm {Pr}_\theta(X = x) can therefore represent the probability of a certain state among all possible states allowed by the state variables. The pseudo-log-likelihood is a similar measure derived from the above expression, namely (in discrete case) l(\theta):=\log L(\theta) = \sum_i \log \mathrm{Pr}_\theta(X_i = x_i\mid X_j = x_j \text{ for } j \neq i). One use of the pseudolikelihood measure is as an approximation for inference about a Markov or Bayesian network, as the pseudolikelihood of an assignment to X_i may often be computed more efficiently than the likelihood, particularly when the latter may require marginalization over a large number of variables.

Properties

Use of the pseudolikelihood in place of the true likelihood function in a maximum likelihood analysis can lead to good estimates, but a straightforward application of the usual likelihood techniques to derive information about estimation uncertainty, or for significance testing, would in general be incorrect.

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